Regular dessins d'enfants with dicyclic group of automorphisms

TL;DR

This paper classifies regular dessins d'enfants with automorphism groups being dicyclic groups, revealing unique structures depending on the parity of n and analyzing their geometric and algebraic properties.

Contribution

It provides a complete classification of regular dessins d'enfants with dicyclic automorphism groups, detailing their number, construction on hyperelliptic Riemann surfaces, and Jacobian decompositions.

Findings

Exactly one such dessin for even n ≥ 2

Exactly two such dessins for odd n ≥ 3

Jacobian varieties have a single isotypical component

Abstract

Let $G_{n}$ be the dicyclic group of order $4n$. We observe that, up to isomorphisms, (i) for $n \geq 2$ even there is exactly one regular dessin d'enfant with automorphism group $G_{n}$, and (ii) for $n \geq 3$ odd there are exactly two of them. All of them are produced on very well known hyperelliptic Riemann surfaces. We observe, for each of these cases, that the isotypical decomposition, induced by the action of $G_{n}$, of its jacobian variety has only one component. If $n$ is even, then the action is purely-non-free, that is, every element acts with fixed points. In the case $n$ odd, the action is not purely-non-free in one of the actions and purely non-free for the other.